Meshless Conservative Scheme for Multivariate Nonlinear Hamiltonian PDEs

摘要

For multivariate nonlinear Hamiltonian equations, we propose a meshless conservative method by using radial basis approximation. Based on the method of lines, we first discretize the Hamiltonian functional using radial basis function interpolation, and then obtain a finite-dimensional semi-discrete Hamiltonian system. Moreover, we define a discrete symplectic form and verify that it is an approximation to the continuous one and is conserved with respect to time. For time discretization, two conservative methods (symplectic method and energy-conserving method) are employed to derive the full-discretized system. Approximation errors together with conservation properties including symplecticity and energy are discussed in detail. Finally, we present several numerical examples to illustrate that our method is accurate and effective when processing nonlinear Hamiltonian equations with scattered nodes. Besides, the numerical results also confirm the excellent conservation properties of the proposed method.